Graph each function using end behavior, intercepts, and completing the square to write the funct

step 1 Okay, so we have a function. We're going to complete the square and graph it, including all of t

Graph each function using end behavior, intercepts, and...

Key Concepts

Quadratic Functions

A quadratic function is typically expressed in the form ax² + bx + c and represents a parabola. The coefficient a determines the opening direction of the parabola (upward if a is positive and downward if a is negative), and the function's symmetry is centered on its vertical axis which passes through the vertex.

End Behavior

End behavior describes how the values of the function behave as x approaches positive or negative infinity. For quadratic functions, if a > 0, both ends of the graph rise, and if a < 0, both ends fall. This concept helps in predicting the shape of the graph outside the immediate area around the vertex.

Intercepts

Intercepts are the points where the graph crosses the axes. The y-intercept is found by evaluating f(0), while the x-intercepts, if they exist, are the solutions to the equation ax² + bx + c = 0. Identifying these intercepts is crucial for accurately sketching the graph of the function.

Completing the Square

Completing the square is a method used to rewrite a quadratic function from its standard form into vertex (or shifted) form, a(x - h)² + k. This form makes it easier to identify the vertex of the parabola and understand the horizontal and vertical translations from the base graph of y = x².

Graph Transformations

Graph transformations in the context of a quadratic function involve shifting, stretching, and reflecting the basic parabola. By rewriting the function in vertex form, one can clearly see the horizontal and vertical shifts, as well as any dilations or reflections relative to the parent graph of y = x².

Quadratic Formula

The quadratic formula, given by x = (-b ± ?(b² - 4ac))/(2a), provides a method for calculating the x-intercepts of a quadratic function by solving the equation ax² + bx + c = 0. It is particularly useful when the quadratic cannot be easily factored.

Transcript

00:01 Okay, so we have a function.

00:02 We're going to complete the square and graph it, including all of the important points, such as the y intercept, x intercepts, and the vertex.

00:12 So to start, we're going to go ahead and complete the square by separating these two terms.

00:16 So, f of x equals, so we want to separate the variable terms from the constant terms.

00:26 So we're going to go ahead to x squared minus 7x, where they're going to.

00:32 Add a constant and in order to complete the square and have that that constant term that we already had in the equation as it's on its own on the right not in the parentheses here so i want to make this so we're going to factor out the two our a term from the left and then we need to find out what our constant will be do this we simply do b over 2 squared so our b is a negative 7 half.

01:13 So negative 7 over 2 divided by 2 squared.

01:19 That's negative 7 over 4 squared, which would be negative 49 over 16.

01:28 Now we know that our constant is negative 49 over 16.

01:35 And we have, you can't forget about our 2 that we fact it out earlier.

01:40 That would mean that we've added 2 times negative 9.

01:44 49 over 16 to the left and we need to cancel that out on the right side by subtracting our 2 times negative 49 over 16 and we can lower our denominator to over 8 by a multiplied by 2 so we can subtract our negative 49 over 8 on the right perfect so now we can go ahead and factor out by completing the square and figure out what the factor is.

02:26 So we have 2 times x and we just take that b value over 2.

02:49 Okay, and there's one mistake that made here.

02:52 When we squared our negative 7 over 4, that would make it positive value.

02:57 So this is a positive 49...

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